where parameter \(\tau \in [0,\infty )\)  controls the strength of group influence. The larger the value of \(\tau \) , the less important the influence of the group is, and vice versa. We deliberately choose a Fermi-like probability function, which is a popular updating protocol in evolutionary game theory (EGT) and its stochasticity is similar to the real-world human decision-making process [1]}, [2]}, [3]}, [4]}. The probability in Eq. (REF ) reflects the popular tendency of individuals to refrain from adopting behaviours that contradict their environment or group norm. \(P_{i\rightarrow j}\)  is symmetric around \(d_{i,\nu _i}=1\) , so that if a node is very aligned with its environment (\(d_{i,\nu _i}\rightarrow 0\) ) it has less probability of copying node \(j\) ; and vice versa, if node \(i\)  is not aligned with its group (\(d_{i,\nu _i}\rightarrow 2\) ) the probability to copy \(j\)  increases.
When \(\tau \rightarrow 0\)  the probability of copying the state of the neighbor tends to a step-function, which leads the system to frozen disordered states. On the other end, when \(\tau \rightarrow \infty \)  the MVM becomes the voter model with a rescaled activation rate that will make the process to evolve slower but eventually to reach consensus. The case where \(0<\tau <\infty \)  is akin to introducing heterogeneous and dynamic activation rates dependant upon the states of node \(i\)  and its group, and leads to system organization into competition between opinion domains.
<FIGURE>